Hilbert's tenth problem for families of Zₚ -extensions of imaginary quadratic fields

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in Zₚ-towers of imaginary quadratic fields K. For a odd prime p, the lines (a, b) P¹ (Zₚ) are identified with Zₚ-extensions K₀, ₁/K. Under certain conditions on K that involve explicit elliptic curves, we identify a line (a₀, b₀) P¹ (Z/pZ) such that for all (a, b) P¹ (Zₚ) with (a, b) (a₀, b₀) p, Hilbert's tenth problem has a negative answer in all finite layers of K₀, ₁. Using results of Kriz--Li and Bhargava et al. , we demonstrate that for primes p = 3, 11, 13, 31, 37, a positive proportion of imaginary quadratic fields meet our criteria.